Sinai was the grandson of mathematician Benjamin F. Kagan, the founding head of the Department of Differential Geometry at Moscow State University (MSU), where Sinai received a bachelor’s degree (1957), a Ph.D. (1960), and a doctorate (1963) and studied under the noted statistician Andrey N. Kolmogorov. Sinai then worked as a scientific researcher (1960–71) at MSU’s Laboratory of Probability and Statistical Methods. In 1968 he signed a letter supporting the poet, mathematician, and longtime dissident Aleksandr Yesenin-Volpin. As a punishment for having signed the petition, Sinai was not allowed to travel abroad for several years. He remained at MSU as a professor and in 1971 took a post as a senior researcher at the Landau Institute for Theoretical Physics of the Russian Academy of Sciences. He left MSU in 1993 to become a professor of mathematics at Princeton University while retaining his position at the Landau Institute.

Sinai’s first notable contribution to mathematics was the development of the Kolmogorov-Sinai entropy. In collaboration with Kolmogorov, he built upon the work of American engineer Claude Shannon, who developed a measure of the efficiency of a communications system, called the entropy, that is computed on the basis of the statistical properties of the message source. (In Shannon’s information theory, the entropy is analogous to the thermodynamic concept of entropy, which measures the amount of disorder in physical systems.) Sinai and Kolmogorov in 1959 extended this idea of entropy to determine a measure of the complexity of the motion taking place in a dynamical system. The Kolmogorov-Sinai entropy proved to be widely applicable; Russian mathematician Yakov Pesin later demonstrated that when the Kolmogorov-Sinai entropy is greater than zero, the dynamical system will display chaos; that is, a slight difference in the initial conditions will produce vastly different trajectories.

Although best known for his contributions to chaos theory, Sinai also did notable work in dynamical billiards, in which the trajectory of a massless point is followed around a “billiards table,” for which the surface can take any shape (even three or more dimensions). The Sinai billiard, which he introduced in 1963, was a flat square with a circle cut out of the middle. Sinai proved that the trajectories of the billiard ball were ergodic; that is, they filled the space between the square and circular walls. The trajectories were also chaotic.